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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Preservation of the residual classes numbers by polynomials


Authors: Jean-Luc Chabert and Youssef Fares
Journal: Proc. Amer. Math. Soc. 139 (2011), 2423-2430
MSC (2010): Primary 11C08; Secondary 11A07, 11R09
Published electronically: December 20, 2010
MathSciNet review: 2784807
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Abstract: Let $ K$ be a global field and let $ \mathcal{O}_{K,S}$ be the ring of $ S$-integers of $ K$ for some finite set $ S$ of primes of $ K$. We prove that whatever the infinite subset $ E\subseteq \mathcal{O}_{K,S}$ and the polynomial $ f(X)\in K[X]$, the subsets $ E$ and $ f(E)$ have the same number of residual classes modulo $ \mathfrak{m}$ for almost all maximal ideals $ \mathfrak{m}$ of $ \mathcal{O}_{K,S}$ if and only if $ \deg(f)=1$ when the characteristic of $ K$ is 0 and $ f(X)=g(X^{p^k})$ for some integer $ k$ and some polynomial $ g$ with $ \deg(g)=1$ when the characteristic of $ K$ is $ p>0$.


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Additional Information

Jean-Luc Chabert
Affiliation: Département de Mathématiques, LAMFA CNRS-UMR 6140, Université de Picardie, 80039 Amiens, France
Email: jean-luc.chabert@u-picardie.fr

Youssef Fares
Affiliation: Département de Mathématiques, LAMFA CNRS-UMR 6140, Université de Picardie, 80039 Amiens, France
Email: youssef.fares@u-picardie.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10696-2
PII: S 0002-9939(2010)10696-2
Keywords: Polynomial mappings, global fields, $S$-integers, $S$-units
Received by editor(s): April 11, 2010
Received by editor(s) in revised form: July 7, 2010
Published electronically: December 20, 2010
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.